What math formulas should I know for ParaPathways?

Perimeter, area, volume, surface area, percent, mean, median, mode and range, plus the conversions inside the US customary and metric systems, are the formulas on this page, each as a table, with three worked examples at the end. An on-screen four-function calculator is provided on the Mathematics subtest.

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What ETS's skills list covers, and the one exam fact

ETS publishes a skills list for the Mathematics subtest, and it covers percents, perimeter, area, surface area, volume, measurement conversion, and the calculation of mean, median, mode and range. Every one of those topics has a table below, written as plain arithmetic with the steps in order. What is on this page is general math: it is not a prediction of what a given version of the test will ask.

One exam fact belongs on this page, and ETS states it plainly: an on-screen four-function calculator is provided on the Mathematics subtest (5759). A four-function calculator adds, subtracts, multiplies and divides. How much you use it, and when, is your call on the day.

Perimeter and area

Perimeter is the distance all the way around a shape. Area is the surface a shape covers. Both use the same measurements, so write down which one the question asks for before you pick a formula.

What you are finding	Formula
Perimeter of a rectangle	2 x (length + width)
Perimeter of a square	4 x side
Perimeter of any polygon	add up every side
Area of a rectangle	length x width
Area of a square	side x side
Area of a triangle	(base x height) / 2

A polygon has no formula of its own: add up every side once and you have the perimeter. A square is the special case of a rectangle where the length and the width are the same number, which is why both rows exist.

Volume and surface area of a rectangular prism

A rectangular prism is a box: three pairs of equal faces, and the measurement you want tells you which pair of faces matters.

What you are finding	Formula
Volume (how much fits inside)	length x width x height
Surface area (how much covers it)	2 x (length x width + length x height + width x height)
Area of one face	length x width

Surface area counts every face of the box, and the shortest way to get there is to find the area of one face of each size and then double the total.

Percent

A percentage is a part out of a hundred, so a percent goes into a formula as a decimal. That first row is the one that decides whether the rest of the work comes out right.

What you are finding	Formula
Turn a percent into a decimal	15% is 0.15; 40% is 0.4
Find the part	part = percent x whole
Find the percent	percent = (part / whole) x 100
Find the whole	whole = part / percent
Find a percent change	(new value - old value) / old value x 100

Skipping that step turns a correct setup into an answer one hundred times too big, so write the decimal down and use it in the next row.

Mean, median, mode and range

These four describe a set of numbers rather than a shape, and they do not all start the same way.

What you are finding	Formula
Mean (the average)	add the values, then divide by how many there are
Median	the middle value in order; with an even count, the mean of the two middle values
Mode	the value that appears most often
Range	largest value - smallest value

Order the values from smallest to largest before you look for a median, and count how often each value appears before you name a mode. A list where every value appears once has no mode, and a list where every value appears the same number of times has no single mode either.

Conversions within US customary units

Every factor in this table stays inside the US customary system, so you multiply or divide and the family of units never changes.

Conversion	The fact to remember
Length: foot and inches	1 foot = 12 inches
Length: yard, foot, inch	1 yard = 3 feet = 36 inches
Length: mile and feet	1 mile = 5,280 feet
Weight: pound and ounces	1 pound = 16 ounces
Weight: ton and pounds	1 ton = 2,000 pounds
Liquid volume: cup and fluid ounces	1 cup = 8 fluid ounces
Liquid volume: pint and cups	1 pint = 2 cups = 16 fluid ounces
Liquid volume: quart and pints	1 quart = 2 pints = 4 cups
Liquid volume: gallon and quarts	1 gallon = 4 quarts = 8 pints = 16 cups
Time: hour and minutes	1 hour = 60 minutes
Time: minute and seconds	1 minute = 60 seconds
Time: day and hours	1 day = 24 hours

Write the unit down at each step, so 3 feet becomes 36 inches and not 36 of something unnamed. Choose the row that matches the unit the question asks for: 3 yards is 9 feet, and it is also 108 inches.

Conversions within the metric system

The metric system is built on powers of ten, so converting inside it means multiplying or dividing by 10, 100 or 1,000.

Conversion	The fact to remember
Length: kilometer and meters	1 kilometer = 1,000 meters
Length: meter and centimeters	1 meter = 100 centimeters
Length: centimeter and millimeters	1 centimeter = 10 millimeters
Length: meter and millimeters	1 meter = 1,000 millimeters
Mass: kilogram and grams	1 kilogram = 1,000 grams
Mass: gram and milligrams	1 gram = 1,000 milligrams
Liquid volume: liter and milliliters	1 liter = 1,000 milliliters
Liquid volume: milliliter and cubic centimeter	1 milliliter = 1 cubic centimeter

Every factor here is a power of ten (10, 100 or 1,000), except the last row, where a milliliter and a cubic centimeter are the same amount of space.

Three worked examples

These are the formulas doing their job, written out step by step. Try each one before you read the steps. They are not practice-test items, and nothing here predicts what a given version of the test will ask.

Worked example 1 - Surface area

A classroom storage bin is 40 cm long, 30 cm wide and 20 cm high. What is its surface area, in square centimeters?

One of each face: 40 x 30 = 1,200; 40 x 20 = 800; 30 x 20 = 600.

Add the three: 1,200 + 800 + 600 = 2,600.

Double it, because a box has two faces of each size: 2 x 2,600 = 5,200.

Units: lengths in centimeters give an area in square centimeters.

Answer: 5,200 square centimeters.

Worked example 2 - Percent change

A school district bought 250 tablets last year and 300 this year. What is the percent change from last year to this year?

Change: 300 - 250 = 50.

Divide by the old value, the number you started from: 50 / 250 = 0.2.

Turn it into a percent: 0.2 x 100 = 20%.

The district bought more tablets, so it is an increase.

Answer: a 20% increase.

Worked example 3 - Unit conversion

A classroom has 3 liters of water and each bottle holds 250 milliliters. How many bottles can be filled?

Go to the smaller unit first: 3 liters x 1,000 = 3,000 milliliters.

Divide by what one bottle holds: 3,000 / 250 = 12.

Units line up: milliliters divided by milliliters leaves a count of bottles.